Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{5q^2 + 55q + 150}{-3q^3 + 12q^2 + 135q}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {5(q^2 + 11q + 30)} {-3q(q^2 - 4q - 45)} $ $ r = -\dfrac{5}{3q} \cdot \dfrac{q^2 + 11q + 30}{q^2 - 4q - 45} $ Next factor the numerator and denominator. $ r = - \dfrac{5}{3q} \cdot \dfrac{(q + 5)(q + 6)}{(q + 5)(q - 9)}$ Assuming $q \neq -5$ , we can cancel the $q + 5$ $ r = - \dfrac{5}{3q} \cdot \dfrac{q + 6}{q - 9}$ Therefore: $ r = \dfrac{ -5(q + 6)}{ 3q(q - 9)}$, $q \neq -5$